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<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.st_eg.two_sample_students_t"></a><a class="link" href="two_sample_students_t.html" title="Comparing the means of two samples with the Students-t test">Comparing
          the means of two samples with the Students-t test</a>
</h5></div></div></div>
<p>
            Imagine that we have two samples, and we wish to determine whether their
            means are different or not. This situation often arises when determining
            whether a new process or treatment is better than an old one.
          </p>
<p>
            In this example, we'll be using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm" target="_top">Car
            Mileage sample data</a> from the <a href="http://www.itl.nist.gov" target="_top">NIST
            website</a>. The data compares miles per gallon of US cars with miles
            per gallon of Japanese cars.
          </p>
<p>
            The sample code is in <a href="../../../../../../example/students_t_two_samples.cpp" target="_top">students_t_two_samples.cpp</a>.
          </p>
<p>
            There are two ways in which this test can be conducted: we can assume
            that the true standard deviations of the two samples are equal or not.
            If the standard deviations are assumed to be equal, then the calculation
            of the t-statistic is greatly simplified, so we'll examine that case
            first. In real life we should verify whether this assumption is valid
            with a Chi-Squared test for equal variances.
          </p>
<p>
            We begin by defining a procedure that will conduct our test assuming
            equal variances:
          </p>
<pre class="programlisting"><span class="comment">// Needed headers:</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="comment">// Simplify usage:</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>

<span class="keyword">void</span> <span class="identifier">two_samples_t_test_equal_sd</span><span class="special">(</span>
        <span class="keyword">double</span> <span class="identifier">Sm1</span><span class="special">,</span>       <span class="comment">// Sm1 = Sample 1 Mean.</span>
        <span class="keyword">double</span> <span class="identifier">Sd1</span><span class="special">,</span>       <span class="comment">// Sd1 = Sample 1 Standard Deviation.</span>
        <span class="keyword">unsigned</span> <span class="identifier">Sn1</span><span class="special">,</span>     <span class="comment">// Sn1 = Sample 1 Size.</span>
        <span class="keyword">double</span> <span class="identifier">Sm2</span><span class="special">,</span>       <span class="comment">// Sm2 = Sample 2 Mean.</span>
        <span class="keyword">double</span> <span class="identifier">Sd2</span><span class="special">,</span>       <span class="comment">// Sd2 = Sample 2 Standard Deviation.</span>
        <span class="keyword">unsigned</span> <span class="identifier">Sn2</span><span class="special">,</span>     <span class="comment">// Sn2 = Sample 2 Size.</span>
        <span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span>     <span class="comment">// alpha = Significance Level.</span>
<span class="special">{</span>
</pre>
<p>
            Our procedure will begin by calculating the t-statistic, assuming equal
            variances the needed formulae are:
          </p>
<div class="blockquote"><blockquote class="blockquote"><p>
              <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial1.svg"></span>

            </p></blockquote></div>
<p>
            where Sp is the "pooled" standard deviation of the two samples,
            and <span class="emphasis"><em>v</em></span> is the number of degrees of freedom of the
            two combined samples. We can now write the code to calculate the t-statistic:
          </p>
<pre class="programlisting"><span class="comment">// Degrees of freedom:</span>
<span class="keyword">double</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sn2</span> <span class="special">-</span> <span class="number">2</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Degrees of Freedom"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// Pooled variance:</span>
<span class="keyword">double</span> <span class="identifier">sp</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(((</span><span class="identifier">Sn1</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">Sn2</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">v</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Pooled Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sp</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// t-statistic:</span>
<span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Sm1</span> <span class="special">-</span> <span class="identifier">Sm2</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">sp</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="number">1.0</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="number">1.0</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">));</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"T Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
            The next step is to define our distribution object, and calculate the
            complement of the probability:
          </p>
<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">q</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t_stat</span><span class="special">)));</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability that difference is due to chance"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
   <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">q</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
            Here we've used the absolute value of the t-statistic, because we initially
            want to know simply whether there is a difference or not (a two-sided
            test). However, we can also test whether the mean of the second sample
            is greater or is less (one-sided test) than that of the first: all the
            possible tests are summed up in the following table:
          </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                    <p>
                      Hypothesis
                    </p>
                  </th>
<th>
                    <p>
                      Test
                    </p>
                  </th>
</tr></thead>
<tbody>
<tr>
<td>
                    <p>
                      The Null-hypothesis: there is <span class="bold"><strong>no difference</strong></span>
                      in means
                    </p>
                  </td>
<td>
                    <p>
                      Reject if complement of CDF for |t| &lt; significance level
                      / 2:
                    </p>
                    <p>
                      <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
                      <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
                      <span class="special">&lt;</span> <span class="identifier">alpha</span>
                      <span class="special">/</span> <span class="number">2</span></code>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      The Alternative-hypothesis: there is a <span class="bold"><strong>difference</strong></span>
                      in means
                    </p>
                  </td>
<td>
                    <p>
                      Reject if complement of CDF for |t| &gt; significance level
                      / 2:
                    </p>
                    <p>
                      <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
                      <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
                      <span class="special">&gt;</span> <span class="identifier">alpha</span>
                      <span class="special">/</span> <span class="number">2</span></code>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>less</strong></span>
                      than Sample 2 Mean.
                    </p>
                  </td>
<td>
                    <p>
                      Reject if CDF of t &gt; significance level:
                    </p>
                    <p>
                      <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
                      <span class="identifier">t</span><span class="special">)</span>
                      <span class="special">&gt;</span> <span class="identifier">alpha</span></code>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>greater</strong></span>
                      than Sample 2 Mean.
                    </p>
                  </td>
<td>
                    <p>
                      Reject if complement of CDF of t &gt; significance level:
                    </p>
                    <p>
                      <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
                      <span class="identifier">t</span><span class="special">))</span>
                      <span class="special">&gt;</span> <span class="identifier">alpha</span></code>
                    </p>
                  </td>
</tr>
</tbody>
</table></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
              For a two-sided test we must compare against alpha / 2 and not alpha.
            </p></td></tr>
</table></div>
<p>
            Most of the rest of the sample program is pretty-printing, so we'll skip
            over that, and take a look at the sample output for alpha=0.05 (a 95%
            probability level). For comparison the dataplot output for the same data
            is in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm" target="_top">section
            1.3.5.3</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
            e-Handbook of Statistical Methods.</a>.
          </p>
<pre class="programlisting">   ________________________________________________
   Student t test for two samples (equal variances)
   ________________________________________________

   Number of Observations (Sample 1)                      =  249
   Sample 1 Mean                                          =  20.145
   Sample 1 Standard Deviation                            =  6.4147
   Number of Observations (Sample 2)                      =  79
   Sample 2 Mean                                          =  30.481
   Sample 2 Standard Deviation                            =  6.1077
   Degrees of Freedom                                     =  326
   Pooled Standard Deviation                              =  6.3426
   T Statistic                                            =  -12.621
   Probability that difference is due to chance           =  5.273e-030

   Results for Alternative Hypothesis and alpha           =  0.0500

   Alternative Hypothesis              Conclusion
   Sample 1 Mean != Sample 2 Mean       NOT REJECTED
   Sample 1 Mean &lt;  Sample 2 Mean       NOT REJECTED
   Sample 1 Mean &gt;  Sample 2 Mean       REJECTED
</pre>
<p>
            So with a probability that the difference is due to chance of just 5.273e-030,
            we can safely conclude that there is indeed a difference.
          </p>
<p>
            The tests on the alternative hypothesis show that we must also reject
            the hypothesis that Sample 1 Mean is greater than that for Sample 2:
            in this case Sample 1 represents the miles per gallon for Japanese cars,
            and Sample 2 the miles per gallon for US cars, so we conclude that Japanese
            cars are on average more fuel efficient.
          </p>
<p>
            Now that we have the simple case out of the way, let's look for a moment
            at the more complex one: that the standard deviations of the two samples
            are not equal. In this case the formula for the t-statistic becomes:
          </p>
<div class="blockquote"><blockquote class="blockquote"><p>
              <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial2.svg"></span>

            </p></blockquote></div>
<p>
            And for the combined degrees of freedom we use the <a href="http://en.wikipedia.org/wiki/Welch-Satterthwaite_equation" target="_top">Welch-Satterthwaite</a>
            approximation:
          </p>
<div class="blockquote"><blockquote class="blockquote"><p>
              <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial3.svg"></span>

            </p></blockquote></div>
<p>
            Note that this is one of the rare situations where the degrees-of-freedom
            parameter to the Student's t distribution is a real number, and not an
            integer value.
          </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
              Some statistical packages truncate the effective degrees of freedom
              to an integer value: this may be necessary if you are relying on lookup
              tables, but since our code fully supports non-integer degrees of freedom
              there is no need to truncate in this case. Also note that when the
              degrees of freedom is small then the Welch-Satterthwaite approximation
              may be a significant source of error.
            </p></td></tr>
</table></div>
<p>
            Putting these formulae into code we get:
          </p>
<pre class="programlisting"><span class="comment">// Degrees of freedom:</span>
<span class="keyword">double</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">;</span>
<span class="identifier">v</span> <span class="special">*=</span> <span class="identifier">v</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">t1</span> <span class="special">=</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span><span class="special">;</span>
<span class="identifier">t1</span> <span class="special">*=</span> <span class="identifier">t1</span><span class="special">;</span>
<span class="identifier">t1</span> <span class="special">/=</span>  <span class="special">(</span><span class="identifier">Sn1</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">t2</span> <span class="special">=</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">;</span>
<span class="identifier">t2</span> <span class="special">*=</span> <span class="identifier">t2</span><span class="special">;</span>
<span class="identifier">t2</span> <span class="special">/=</span> <span class="special">(</span><span class="identifier">Sn2</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
<span class="identifier">v</span> <span class="special">/=</span> <span class="special">(</span><span class="identifier">t1</span> <span class="special">+</span> <span class="identifier">t2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Degrees of Freedom"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// t-statistic:</span>
<span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Sm1</span> <span class="special">-</span> <span class="identifier">Sm2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">);</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"T Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
            Thereafter the code and the tests are performed the same as before. Using
            are car mileage data again, here's what the output looks like:
          </p>
<pre class="programlisting">   __________________________________________________
   Student t test for two samples (unequal variances)
   __________________________________________________

   Number of Observations (Sample 1)                      =  249
   Sample 1 Mean                                          =  20.145
   Sample 1 Standard Deviation                            =  6.4147
   Number of Observations (Sample 2)                      =  79
   Sample 2 Mean                                          =  30.481
   Sample 2 Standard Deviation                            =  6.1077
   Degrees of Freedom                                     =  136.87
   T Statistic                                            =  -12.946
   Probability that difference is due to chance           =  1.571e-025

   Results for Alternative Hypothesis and alpha           =  0.0500

   Alternative Hypothesis              Conclusion
   Sample 1 Mean != Sample 2 Mean       NOT REJECTED
   Sample 1 Mean &lt;  Sample 2 Mean       NOT REJECTED
   Sample 1 Mean &gt;  Sample 2 Mean       REJECTED
</pre>
<p>
            This time allowing the variances in the two samples to differ has yielded
            a higher likelihood that the observed difference is down to chance alone
            (1.571e-025 compared to 5.273e-030 when equal variances were assumed).
            However, the conclusion remains the same: US cars are less fuel efficient
            than Japanese models.
          </p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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